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Doesn't this explain it?
type of people that we are worried about appearing prejudiced in front of
No, I don't think it explains much, at all. I think it explains something about the speaker's insecurities. I think that if race is on your mind a lot, that your anxieties will be about race. But if you have some other trauma, then that will be what your anxieties will be about. So I don't think that this has anything to do with race, but with insecurity.
All people of all races have insecurities.
There's a reason people who aren't the majority race often have race top of mind.
Maybe you should ask someone who's experienced it.
Sure. It's useful to consider the difference when the minority status is noticeable only after conversation and intimate contact (many religious ones; sexual ones; cultural ones) vs when it's immediately noticeable to anyone in visual range, but any difference can lead to different affordances in interaction.
Why is race important in this?